The role of autologous marrow transplantation (AMT) in the treatment of patients with acute nonlymphoblastic leukemia (ANL) will be evaluated. A randomized trial will be carried out in cooperation with the Southwest Oncology Group (SWOG) to determine the efficacy and appropriate timing of AMT. Patients with ANL who achieve remission will be HLA typed. Those with matched donors will be offered allogeneic transplantation. Those without matched donors will have marrow cryopreserved following 4-hydroperoxycyclo- phosphamide (4-HC) treatment. They will then be randomized to receive AMT in first remission or to be treated with high dose combination chemotherapy with AMT reserved until first relapse. This will allow for a comparison of: 1) autografts versus allografts in first complete remission and 2) autografts in first complete remission versus chemotherapy with autografting reserved until first relapse. In addition, we will attempt to define more clearly the curative potential of AMI at first relapse by transplanting additional patients at this stage of disease who are not registered on the SWOG study.